What the puzzle asks

A Pythagorean Path puzzle gives you a list of distances in order. Your job is to draw a single continuous path that visits every dot once so that the distance between each pair of consecutive dots equals the next length in the list. The dots lie on a regular grid, so their coordinates are integers.

Key idea: distances on a grid come from the Pythagorean Theorem

If one step moves from (x1,y1) to (x2,y2), and dx = x2 - x1 and dy = y2 - y1, then the distance is:

distance = sqrt(dx^2 + dy^2)

Because x and y are integers, each allowed distance is the square root of an integer. So for a given length like sqrt(5) you must find a step with dx^2 + dy^2 = 5 (for example dx=2, dy=1).

Common small grid moves (dx, dy) and their lengths

• (1, 0) or (0, 1) — length = 1 (sqrt(1))
• (1, 1) — length = sqrt(2)
• (2, 0) — length = 2 (sqrt(4))
• (2, 1) — length = sqrt(5)
• (2, 2) — length = sqrt(8)
• (3, 1) — length = sqrt(10)
• (3, 2) — length = sqrt(13)
• (3, 3) — length = sqrt(18)
• (4, 1) — length = sqrt(17)

Remember: signs (±) and swapping dx,dy give many directions for each length (for example (2,1), (2,-1), (-2,1), (-2,-1), (1,2), etc.).

Step-by-step solving method

1. Write each length as a target pair of squared integers.

   Example: sqrt(5) means dx^2 + dy^2 = 5 → possibilities (2,1) or (1,2) in some sign/direction.

2. List all possible (dx,dy) moves for each length.

   Make small sketches or notes showing all directions for those vectors.

3. Pick a starting dot and try to place moves in order.

   Use pencil marks: when you place one move you physically check whether the next move can be made from the new dot without leaving the grid or revisiting a dot.

4. Use forced moves to prune possibilities.

   Some dots (like corners or edges) will have fewer possible outgoing moves. If a dot must be used in a particular way to satisfy the next lengths, that forces choices and reduces the branches you must try.

5. Backtrack when you hit a dead end.

   If following a sequence prevents you from placing a later length, undo the last choices and try a different direction. Most puzzles are solved by careful backtracking with logical pruning.

Extra strategies that help

• Spot unique endpoints: The first and last dots are endpoints (degree 1); other dots in the path have degree 2. If a dot can only be the end of the path because no two moves can meet there, it's likely an endpoint.
• Use symmetry: If the grid and lengths are symmetric, you can try one half and mirror it, or reduce duplicate cases.
• Count available moves: For each remaining dot, count how many of the remaining move-vectors can reach it. Dots with very few options are where you should focus first.
• Mark forbidden dots: When you decide a dot is already used or will be unreachable, mark it so you don’t waste time trying to return to it.

A short worked example

Suppose you have a small 3×3 grid with coordinates (0..2, 0..2) and the length list: sqrt(5), sqrt(2), 1. Can we connect four dots?

1. sqrt(5) → move (2,1) or (1,2) in some direction. Try starting at (0,0) and use (2,1) to go to (2,1).
2. sqrt(2) → from (2,1) use (-1,1) to go to (1,2) because (-1)^2 + 1^2 = 2.
3. 1 → from (1,2) go down (0,-1) to (1,1).

The path of dots is (0,0) → (2,1) → (1,2) → (1,1). Each step matches the given lengths and no dot is visited twice. This shows how to translate lengths into concrete grid moves and test them.

Final tips

• Make a list of possible (dx,dy) for every length before you try to place anything.
• Start from likely endpoints (corners or edge dots) because endpoints reduce branching.
• Work systematically and don’t be afraid to erase and backtrack — that’s part of the process.

Practice these steps on the puzzles in Beast Academy pages 58–60: list moves, try placements, prune using forced moves, and backtrack when needed. With experience you'll get faster at spotting which moves will fit the grid. Good luck!